Front-end electronics for capacitive sensors typically includes a preamplifier followed by a filter. The preamplifier provides low-noise amplification of the signals induced in the sensor electrodes. The filter, by properly limiting the signal bandwidth, maximizes the Signal-to-Noise (S/N) ratio. Additionally the filter limits the duration of the output signal associated with the measured event and, for those sensors where the induced signal is relatively slow, it maximizes the signal amplitude, i.e. it minimizes the ballistic deficit, as described in G. F. Knoll, “Radiation detection and measurement”, 3rd ed., John Wiley & Sons, 2000, which is incorporated by reference in its entirety as if fully set forth in this specification.
Filters can be either time-variant or time-invariant. In electronics for radiation sensors, time-invariant filters are frequently referred to as “shapers” since, in a time-domain view, they “shape” the response associated with events. Filters can also be synthesized digitally, even though in most cases this is impractical due to constraints from power and real-estate budgets.
S/N ratio and dynamic range are important parameters reflecting performance of a shaper. In the following, we analyze classical shapers based on voltage feedback with passive components with respect to noise and dynamic range. Charge amplifiers, along with providing low-noise amplification, offer a low input impedance (i.e., virtual ground) which stabilizes the potential of a sensor electrode and reduces inter-electrode cross-talk. A charge amplifier 100 is schematized in FIG. 1, where we assume an ideal voltage amplifier with infinite gain and bandwidth. A finite gain and bandwidth would have negligible consequence on our analysis, if the dc loop gain is high and the rise time is a small fraction of the peaking time.
The current Ii induced in the sensing electrode is amplified with current gain (or charge gain) Ac equal to the ratio of the feedback impedance Zf and the coupling impedance Zc. This ratio (i.e., the gain Ac) must be a real number, in order to avoid undesired tails in the output current Is injected in the next stage. The output current Is is injected, with opposite polarity, into the next stage, which offers another virtual ground and represents the input stage of a shaper 110. The current (or charge) is then filtered and converted into a voltage Vo with transfer function Zs. It is followed by further processing such as discrimination, peak- or time-detection, and/or counting. It is worth noting that the charge amplifier 100 can be realized using two or more charge-amplification stages with gains Ac1, Ac2, . . . , and overall charge gain Ac is given by the product of those. This is usually done when large values of Ac are required, such as for sensors generating very small signals.
For simplicity we assume for Zf an infinite resistive component and a finite capacitive component Cf. This is justified considering that designers tend to keep the resistive component as high as possible in order to minimize the parallel noise contribution at the front-end. The coupling impedance will be capacitive according to Cc=CfAc. We also assume, initially, that the input stage of the shaper 110 is realized using a transimpedance amplifier with feedback impedance Z1=R1//C1, providing the first pole of the shaper with time constant τ1=R1C1. Finally, we assume that the shaper amplifiers are characterized by infinite gain and are noiseless. The latter is justified by the fact that, in most practical cases, the noise contribution from the amplifiers can be made negligible by increasing the size and power of active devices. If this is not easy to achieve, then the noise from the amplifiers must be taken into account. The configuration resulting from these assumptions is shown in FIG. 2. FIG. 2 is a schematic circuit of a charge amplifier 210 assuming a capacitive feedback for the charge amplifier 210 and a single-pole transimpedance amplifier as an input stage 220 of a shaper 200, where the output waveform V1 in response to a charge Q is also shown, with peak amplitude Q·Ac/C1. The output V1 may be input into a subsequent stage 230 of the shaper 200.
Starting from these assumptions and from the configuration in FIG. 2, we can calculate the contribution to the Equivalent Noise Charge (ENC) of the first stage 220 of the shaper 200. The noise contribution comes from the dissipative component R1 of the shaper, as described in A. F. Arbel, “The second stage noise contribution of a nuclear pulse amplifier”, IEEE Trans. Nucl. Sci., vol. 15, no. 5, pp. 2-5, 1968, which is incorporated by reference in its entirety as if fully set forth in this specification. The parallel noise spectral density of R1 is given by 4kT/R1 and it can be reported as an equivalent parallel noise generator at the input of the charge amplifier 210 by scaling it with the square of the charge gain Ac. It must be kept in mind that this is done for calculation purposes and the actual noise source is further down in the channel, not to be confused with the physical sources of parallel noise at the input. It follows the contribution to the ENC of R1, given by:
                              ENC                      s            ⁢                                                  ⁢            1                    2                =                                            a              p                                      A              c              2                                ⁢                                    4              ⁢              k              ⁢                                                          ⁢              T                                      R              1                                ⁢                      τ            p                                              (        1        )            where ap is the ENC coefficient for white parallel noise, as described in V. Radeka, “Low noise techniques in detectors”, Ann. Rev. Nucl. Part. Sci., vol. 38, pp. 217-277, 1988, and in E. Gatti and P. F. Manfredi, “Processing the signals from solid state detectors in elementary particle physics”, La Rivista del Nuovo Cimento, vol. 9, pp. 1-147, 1986, and in V. Radeka, “Signal processing for particle detectors”, H. Schopper editor, Landolt-Bornstein, New Series I/21B1, in press, each of which is incorporated by reference in its entirety as if fully set forth in this specification, and τp is the peaking time (1% to peak) of the shaped signal. It is worth noting that an analysis based on front-end amplifier without feedback would give Ac dependent on the input capacitance, as described, for example, in FIG. 6.11 of V. Radeka, “Signal processing for particle detectors”, H. Schopper editor, Landolt-Bornstein, New Series I/21B1, in press, which is not the case for the charge amplifier configuration in FIG. 2. From Equation (1) it can be observed that the contribution decreases as Ac increases. Since the peaking time is proportional to the time constant, τp=ηpR1C1, we can write:
                              ENC                      s            ⁢                                                  ⁢            1                    2                =                                                            a                p                                            A                c                2                                      ⁢                                          4                ⁢                k                ⁢                                                                  ⁢                T                                            R                1                                      ⁢                          η              p                        ⁢                          R              1                        ⁢                          C              1                                =                                                                      a                  p                                ⁢                                  η                  p                                                            A                c                2                                      ⁢            4            ⁢                                                  ⁢            k            ⁢                                                  ⁢            T            ⁢                                                  ⁢                          C              1                                                          (        2        )            where ηp depends on the type of shaping. Table I includes coefficients for unipolar shapers with real (R) and complex-conjugate (C) poles in different orders. Table I summarizes the values of ap and ηp for semi-Gaussian shapers with real poles (even and odd) and complex conjugate poles (odd only) where the input stage is the real pole. Also included in Table I are the coefficient χ, which takes into account the noise contribution of the next stages, and the Relative Dynamic Range (RDR). In Table I are also reported aw, i.e., the ENC coefficient for white series noise, and the coefficients χ and RDR, which will be described later in this application.
TABLE IRU-2RU-3RU-4RU-5RU-6RU-7CU-2CU-3CU-4CU-5CU-6CU-7aw0.920.820.850.890.920.940.930.850.910.961.011.04ap0.920.660.570.520.480.460.880.610.510.460.420.40ηp11.922.743.474.134.73n/a1.79n/a2.95n/a3.76ap ηp0.921.271.561.811.982.18n/a1.09n/a1.355n/a1.503χ11.131.241.311.371.43n/a5.5n/a5.5n/a5.5RDR10.820.720.660.620.59n/a0.51n/a0.45n/a0.43
It can be observed that, for a given shaper and charge gain Ac, the contribution ENCs1 is defined once the value of C1 is defined. The values of Ac and C1 also define, together, the maximum charge Qmax that the linear front-end can process. If V1max is the maximum voltage swing at the output of the stage, it follows:
                                          Q            max                    ⁢                      A            c                          =                                            C              1                        ⁢                          V                              1                ⁢                                                                  ⁢                max                                      ⁢                                                  ⁢            or            ⁢                                                  ⁢                          A              c                                =                                                    C                1                            ⁢                              V                                  1                  ⁢                  max                                                                    Q              max                                                          (        3        )            
We now express the dynamic range DR of the front-end as the ratio between the maximum charge Qmax and the total ENC, which includes the ENCca from the charge amplifier and the ENCs1 from the first stage of the shaper:
                    DR        =                              Q            max                                                              ENC                ca                2                            +                              ENC                                  s                  ⁢                                                                          ⁢                  1                                2                                                                        (        4        )            
A design which aims at offering the highest possible resolution, i.e., lowest possible ENC tends to keep ENCs1 negligible with respect to ENCca. Assuming about 10% (in power) it follows:
                              DR          ≈                                    Q              max                                                      11                ·                                  ENC                                      s                    ⁢                                                                                  ⁢                    1                                    2                                                                    =                                                                              C                  1                                ⁢                                  V                  max                                                            A                c                                                                                      11                  ·                                                                                    a                        p                                            ⁢                                              η                        p                                                                                    A                      c                      2                                                                      ⁢                4                ⁢                k                ⁢                                                                  ⁢                T                ⁢                                                                  ⁢                                  C                  1                                                              =                                    V                              1                ⁢                                                                  ⁢                max                                                                                      11                  ·                                      a                    p                                                  ⁢                                  η                  p                                ⁢                                                      4                    ⁢                    k                    ⁢                                                                                  ⁢                    T                                                        C                    1                                                                                                          (        5        )            
It is important to observe that ENCca depends inherently on the total capacitance CIN at the input of the system in FIG. 1 and on the peaking time τp. Here we assume that the charge amplifier has been already optimized for given CIN and τp, and that the design of the shaper (with the 10% requirement) follows from that. Equation (5) shows that the dynamic range increases with V1max and with the square root of C1. For a given shaper and capacitor value C1 the dynamic is maximized if V1max=Vdd, where Vdd is the maximum voltage allowed by the technology, which means that the shaper amplifier must implement a rail-to-rail output stage. Further increases can only be achieved by increasing the value of C1, which also means increasing Ac as shown in Equation (3) and the area (and power) of the first stage of the shaper. For example, for a 0.13 μm technology with 1.2 V supply and typical Metal-Insulator-Metal (MIM) capacitance of 2 fF/μm2, assuming a CU-3 shaper (apηp=1.09) with available area 30×30 μm, the dynamic range is limited, according to Equation (5), to DR<3,600.
For a given C1, higher values of dynamic range can only be obtained at the expense of the ENC, and the maximum would be achieved when ENCs1 dominates over ENCca. Equation (5) can be written in the more general form:
                                                        DR              ≈                                                V                  dd                                                                                            ρ                      ·                                              a                        p                                                              ⁢                                          η                      p                                        ⁢                                                                  4                        ⁢                        k                        ⁢                                                                                                  ⁢                        T                                                                    C                        1                                                                                                                                                    ρ              =                                                ENC                  2                                                  ENC                                      s                    ⁢                                                                                  ⁢                    1                                    2                                                                                        (        6        )            where ρ>1 is the ratio between the squares of the total ENC and the ENCs1 from the first stage of the shaper.
FIG. 3 shows an example of a compromise between ENC and Dynamic Range (DR) for the circuit in FIG. 2 and ENCca=200 e−. Curve 310 shows how DR varies with charge gain Ac, and curve 320 shows how ENC varies with charge gain Ac. The four cases of ρ=1.1, 2, 11, and 30 are shown, and it can be observed how the dynamic range can be increased at the expense of the ENC. Values of ρ lower than 1.1 (i.e., ENC is dominated by ENCs1) would not benefit the DR but would further limit the resolution by increasing the total ENC. The extreme case is for ρ=1 (i.e., no charge amplification) where DR≈15,000 and ENC≈900 e−. On the other hand, values of ρ higher than 11 (i.e., ENC is dominated by ENCca) would not benefit much the ENC but would further limit the DR.
It is worth emphasizing one more time that the ENCca is assumed defined and optimized for noise (i.e. the charge amplifier is designed for given CIN and τp) and that the design of the shaper follows from that. From Equation (6) it can also be observed that such defined DR does not depend on the peaking time τp. However, once the system is designed with a given optimized ENCca and a given ρ, an adjustment of the peaking time (obtained by scaling the value of the resistors) would in most cases change ENCca and then would modify ρ and DR, while the noise contribution from the shaper would not change.
So far we have assumed as negligible the noise contribution from subsequent stages, which provide the additional poles of the shaper. We first consider the case of real coincident poles. These configurations are frequently referred to as “CR-RCn-1 shapers” since they can be realized using one CR filter followed by n−1 filters of RC type, and they are assumed to be connected at the voltage output of the charge amplifier. The resulting transfer function provides one zero in the origin, which compensates for the pole in the origin from the feedback capacitor of the charge amplifier, and n poles with time constant RC. The order of the shaping is equal to n with zeroes cancelled, and n poles in total. The lowest possible order without divergence of noise is n=2, which results in the well-known and widely adopted CR-RC shaper. The equations in the frequency (Laplace) and time domains are as follows:
                                          H            ⁡                          (              s              )                                =                      1                                          (                                  s                  +                  p                                )                            n                                      ,                              h            ⁡                          (              t              )                                =                                                    1                                                      (                                          n                      -                      1                                        )                                    !                                            ⁢                              t                                  n                  -                  1                                            ⁢                              exp                ⁡                                  (                                      -                    tp                                    )                                            ⁢                                                          ⁢              n                        =            2                          ,        3        ,        4        ,        …                            (        7        )            where n is the order and p is the real pole, coincident.
FIG. 4 shows a frequently adopted configuration for CR-RCn-1 shapers. Each additional i pole is obtained by adding one stage with components Ci, Ri, and Ri/Avi, where Avi is the dc voltage gain. Assuming that the first filter stage (or first stage) 410 of a shaper 400 operates rail-to-rail, as required to minimize its and the following noise contributions, the performance of the shaper 400 is maximized when also the next stages also operate rail-to-rail. This condition is obtained with Av2≈e, Av3≈e/2, Av4≈e/2.25, Av5≈e/2.36, and so on. Next, the noise contribution of the two dissipative components of the second filter stage (or second stage) 420, i.e. R2 and R2/Av2 is evaluated. When reported as equivalent parallel generators at the input of the first pole, the noise spectral densities are respectively given by:
                              S                      R            ⁢                                                  ⁢            2                          =                                                            4                ⁢                k                ⁢                                                                  ⁢                T                                            R                2                                      ⁢                                          R                2                2                                            A                                  v                  ⁢                                                                          ⁢                  2                                2                                      ⁢                                          1                +                                                      ω                    2                                    ⁢                                      R                    1                    2                                    ⁢                                      C                    1                    2                                                                              R                1                2                                              =                                                    4                ⁢                k                ⁢                                                                  ⁢                T                                            R                1                                      ⁢                          1                              A                                  v                  ⁢                                                                          ⁢                  2                                2                                      ⁢                                          R                2                                            R                1                                      ⁢                          (                              1                +                                                      ω                    2                                    ⁢                                      R                    1                    2                                    ⁢                                      C                    1                    2                                                              )                                                          (                  8          ⁢          a                )                                          S                      R            ⁢                                                  ⁢                          2              /              Av                        ⁢                                                  ⁢            2                          =                              4            ⁢            k            ⁢                                                  ⁢            T            ⁢                                                  ⁢                                          R                2                                            A                                  v                  ⁢                                                                          ⁢                  2                                                      ⁢                                          1                +                                                      ω                    2                                    ⁢                                      R                    1                    2                                    ⁢                                      C                    1                    2                                                                              R                1                2                                              =                                                    4                ⁢                k                ⁢                                                                  ⁢                T                                            R                1                                      ⁢                          1                              A                                  v                  ⁢                                                                          ⁢                  2                                                      ⁢                                          R                2                                            R                1                                      ⁢                          (                              1                +                                                      ω                    2                                    ⁢                                      R                    1                    2                                    ⁢                                      C                    1                    2                                                              )                                                          (                  8          ⁢          b                )            and they can be combined as a single noise generator:
                              S          2                =                                            4              ⁢              k              ⁢                                                          ⁢              T                                      R              1                                ⁢                                    C              1                                      C              2                                ⁢                      1                          A                              v                ⁢                                                                  ⁢                2                                              ⁢                      (                          1              +                              1                                  A                                      v                    ⁢                                                                                  ⁢                    2                                                                        )                    ⁢                      (                          1              +                                                ω                  2                                ⁢                                  R                  1                  2                                ⁢                                  C                  1                  2                                                      )                                              (        9        )            where we used R1C1=R2C2 for coincident poles. A contribution such as this can be reported as an equivalent parallel noise generator at the input of a charge amplifier 430 by scaling it with the square of the charge gain Ac. After a few transformations it follows the contribution to the ENC of the second stage 420, given by:
                              ENC                      s            ⁢                                                  ⁢            2                    2                =                                                                              a                  p                                ⁢                                  η                  p                                                            A                c                2                                      ⁢            4            ⁢            k            ⁢                                                  ⁢            T            ⁢                                                  ⁢                          C              1                        ⁢                                          C                1                                            C                2                                      ⁢                          1                              A                                  v                  ⁢                                                                          ⁢                  2                                                      ⁢                          (                              1                +                                  1                                      A                                          v                      ⁢                                                                                          ⁢                      2                                                                                  )                        ⁢                          (                              1                +                                                      a                    w                                                                              η                      p                      2                                        ⁢                                          a                      p                                                                                  )                                =                                    ENC                              s                ⁢                                                                  ⁢                1                            2                        ⁢                                          C                1                                            C                2                                      ⁢                          χ              2                                                          (        10        )            where aw is the ENC coefficient for white parallel noise and χ2 depends on the order of the shaper with χ2≈1 for the second order, 0.83 for the third order, 0.78 for the fourth order, and so on. From Equation (10) it can be observed that the noise contribution from the second stage 420 of the shaper 400, relative to the first, decreases as the order increases and as the C2/C1 ratio increases, and in principle can be made negligible for C2>>C1, i.e. at expenses of area and power.
As the order increases, the noise contribution from the next stages must be added. Eventually, the total contribution from the shaper can be written as:
                              ENC          s          2                =                              ENC                          s              ⁢                                                          ⁢              1                        2                    ⁡                      (                          1              +                              χ                ⁢                                                                  ⁢                                                      C                    1                                                        C                    s                                                                        )                                              (        11        )            where we assume rail-to-rail operation, Cs is the average capacitance per pole, and χ≈1 for the second order, 1.13 for the third order, 1.24 for the fourth order, 1.3 for the fifth order, and so on. It is worth emphasizing that the contribution of each additional stage can be made negligible by increasing its capacitance relative to C1, which at equal gain (rail-to-rail operation) corresponds to a reduction in the value of the resistors.
Next consider the case of complex conjugate poles. These configurations, introduced by Ohkawa as described in A. Ohkawa, M. Yoshizawa, and K. Husimi, “Direct synthesis of the Gaussian filter for nuclear pulse amplifiers”, Nucl. Instrum. & Meth., 138 (1979) 85-92, which is incorporated by reference in its entirety as if fully set forth in this specification, have the advantage of a faster return to zero at equal peaking time with respect to the real poles of the same order. The transfer functions in the frequency (Laplace) domain are:
                                          H            ⁡                          (              s              )                                =                      1                                          (                                  s                  +                                      p                    1                                                  )                            ⁢                                                ∏                                      i                    =                    2                                                                              (                                              n                        +                        1                                            )                                        ⁢                                          /                                        ⁢                    2                                                  ⁢                                                                  ⁢                                  [                                                                                    (                                                  s                          +                                                      r                            i                                                                          )                                            2                                        +                                          i                      i                      2                                                        ]                                                                    ⁢                                  ⁢                              n            =            3                    ,          5          ,          7          ,          …                ⁢                                  ⁢                              H            ⁡                          (              s              )                                =                      1                                          ∏                                  i                  =                  1                                                  n                  ⁢                                      /                                    ⁢                  2                                            ⁢                                                          ⁢                              [                                                                            (                                              s                        +                                                  r                          1                                                                    )                                        2                                    +                                      i                    i                    2                                                  ]                                                    ⁢                                  ⁢                              n            =            2                    ,          4          ,          6          ,          …                                    (        12        )            where n is the order, p1 is the real pole, and rj, ij are the real and imaginary parts of the complex-conjugate poles, obtained as roots of the equation
                    1                  0          !                    -                        s          2                          1          !                    +                        s          4                          2          !                    -                        s          6                          3          !                    +              …        ⁢                                  ⁢                              s                          2              ⁢              n                                            n            !                                =    0    ,while in the time domain the transfer functions are:
                                          s            ⁡                          (              t              )                                =                                                    K                1                            ⁢              exp              ⁢                                                          ⁢                              (                                  -                                      tp                    1                                                  )                                      +                                          ∑                                  i                  =                  2                                                                      (                                          n                      +                      1                                        )                                    ⁢                                      /                                    ⁢                  2                                            ⁢                                                          ⁢                              2                ⁢                                                                        K                    i                                                                    ⁢                exp                ⁢                                                                  ⁢                                  (                                      -                                          tr                      i                                                        )                                ⁢                cos                ⁢                                                                  ⁢                                  (                                                            -                                              ti                        i                                                              +                                          ∠                      ⁢                                                                                          ⁢                                              K                        i                                                                              )                                                                    ⁢                                  ⁢                              n            =            3                    ,          5          ,          7          ,          …                ⁢                                  ⁢                              s            ⁡                          (              t              )                                =                                    ∑                              i                =                1                                            n                ⁢                                  /                                ⁢                2                                      ⁢                                                  ⁢                          2              ⁢                                                                K                  i                                                            ⁢              exp              ⁢                                                          ⁢                              (                                  -                                      tr                    i                                                  )                            ⁢              cos              ⁢                                                          ⁢                              (                                                      -                                          ti                      i                                                        +                                      ∠                    ⁢                                                                                  ⁢                                          K                      i                                                                      )                                                    ⁢                                  ⁢                              n            =            2                    ,          4          ,          6          ,          …                                    (        13        )            where the coefficients Ki (with magnitude |Ki| and argument ∠Ki) are given by:
                                                        K              i                        =                                                            1                                                                                    [                                                                              -                                                          r                              i                                                                                -                                                      ji                            i                                                    +                                                      p                            1                                                                          ]                                            ⁢                                                                        ∏                                                                                    k                              =                              2                                                        ,                                                          k                              ≠                              i                                                                                                                                          (                                                              n                                +                                1                                                            )                                                        ⁢                                                          /                                                        ⁢                            2                                                                          ⁢                                                                                                  ⁢                                                  [                                                                                    -                                                              r                                i                                                                                      -                                                          ji                              i                                                        +                                                          r                              k                              2                                                        +                                                          i                              k                              2                                                                                ]                                                                                      -                                          2                      ⁢                                              ji                        i                                                                                            ⁢                i                            >              1                                ,                                          ⁢                      n            =            3                    ,          5          ,          7          ,          …                ⁢                                  ⁢                              K            i                    =                      1                                                            ∏                                                            k                      =                      1                                        ,                                          k                      ≠                      i                                                                            n                    ⁢                                          /                                        ⁢                    2                                                  ⁢                                                                  ⁢                                  [                                                            -                                              r                        i                                                              -                                          ji                      i                                        +                                          r                      k                      2                                        +                                          i                      k                      2                                                        ]                                            -                              2                ⁢                                  ji                  i                                                                    ⁢                                  ⁢                              n            =            2                    ,          4          ,          6          ,          …                                    (        14        )            
FIG. 5 shows a frequently adopted configuration for these shapers, which includes a charge amplifier 510, a first filter stage 520 and second filter stage 530 of a shaper 500. If n is the order (odd in these cases), the real pole is given by the first filter stage and the complex conjugate poles are given by the (n−1)/2 additional filter stages. Each additional filter stage has a transfer function:
                              H          ⁡                      (            s            )                          =                              -                                          A                vi                                                                                  s                    2                                    ⁢                                      R                    i                                    ⁢                                      C                    i                                    ⁢                                      R                    ia                                    ⁢                                      C                    ia                                                  +                                                      sC                    i                                    ⁡                                      [                                                                  R                        i                                            +                                                                        R                          ia                                                ⁡                                                  (                                                      1                            +                                                          A                              vi                                                                                )                                                                                      ]                                                  +                1                                              =                                    A              vi                                                                        s                  2                                                  ω                  i                  2                                            +                              s                                                      ω                    i                                    ⁢                                      Q                    i                                                              +              1                                                          (        15        )            where the values of ω0=1/τ0 (real pole), ωi and Qi, normalized to the peaking time τp, can be obtained from Table II, which includes design coefficients for unipolar shapers with complex-conjugate poles in different orders. The value of Cia is about 20% of the value of Ci, and we can thus assume an average capacitance per pole Cs≈(Ci+Cia)/2.
Evaluating the noise contribution of the dissipative components of these stages is cumbersome. Eventually, the total contribution from the shaper 500 can be written again as in Equation (11), where we assume again rail-to-rail operation, Cs is the average capacitance per pole, and χ≈5.5 for all orders. In these configurations most of the noise contribution comes from the series resistors Ria. Once again it is worth emphasizing that, apart from the first stage 520, the contribution can be made negligible by increasing the value of the average capacitance per pole Cs. Table I includes the value of χ for various orders.
TABLE IIω0τpω1τpQ1ω2τpQ2ω3τpQ3CU-2—1.0310.541————CU-31.7931.9760.606————CU-4—2.4710.5142.8120.672——CU-52.9453.0660.5433.5320.736——CU-6—3.4000.5073.6120.5764.1780.797CU-73.7583.8420.5234.1280.614.7750.855
Also included in Table I is the relative dynamic range RDR, i.e. the DR normalized to the one for the RU-2 case, assuming the same values for ENCca and ρ (e.g. ρ=11 which is the practical case where ENCs=ENCca/10), and all shapers using the same value of C1 and Cs=C1. From Table I it may appear that low order shapers offer a higher DR. A thorough comparative analysis, though, should include the impact of the shaper on ENCca. For example, under constraint of finite pulse width (e.g. rate constraint) and white dominant series noise, higher order shapers offer a lower ENCca due to the higher symmetry (i.e. longer peaking time at equal width).
Applying these results to Equations (5) and (6) for the dynamic range, we obtain:
                                          DR            ≈                                          Q                max                                                              ρ                  ·                                      ENC                    s                    2                                                                                =                                    V                              1                ⁢                max                                                                                      ρ                  ·                                      a                    p                                                  ⁢                                  η                  p                                ⁢                4                ⁢                                  kT                  ⁡                                      (                                                                  1                                                  C                          1                                                                    +                                              𝒳                                                  C                          s                                                                                      )                                                                                      ⁢                                  ⁢                  ρ          =                                    ENC              2                                      ENC              s              2                                                          (        16        )            
For a given total capacitance CT=C1+(n−1) Cs, where n is the order of the shaper, the DR in (16) has a maximum around:
                                          C            1                    =                                    C              T                                      1              +                                                𝒳                  ⁡                                      (                                          n                      -                      1                                        )                                                                                      ⁢                                  ⁢                  n          =                      shaper            ⁢                                                  ⁢            order                                              (        17        )            which for all of the low order shapers and high order shapers with complex conjugate poles is around CT/n while for high order shapers with real coincident poles is somewhat lower. The rest of the capacitance can be distributed in equal amount among the additional poles, but it should be observed that slightly better results can be obtained by assigning larger capacitance values to the last stage. Since the maximum is relatively shallow, the value of DR obtained for CT/n is still a good approximation.
In the previously reported example with Cs=C1=1.8 pF, ENCca=200 e−, and ρ=11, it follows DR<1,400 and 2,800 for CU-3 and RU-2, respectively. With the described configurations and assuming comparable area and power, the shapers with real poles offer a dynamic range of about 70% higher than the ones with complex conjugate poles.
FIGS. 6(a)-(d) illustrate some configurations of classical shapers which could provide an alternative solution to the voltage feedback circuit in FIG. 2 for realizing a low-noise single pole stage. They make use of CMOS current mirrors to scale down the current in resistor R, thus reducing its noise contribution. In fact, if λ=λ1λ2 is the scaling factor, the dissipative current feedback IF through MF can be approximated as IF≈Vo/Req, where Req=R·λ, is the equivalent resistance, which sets with C the time constant of the filter, given by C·Req. The parallel noise contribution from R, reported at the input, scales down with λ2, being given by 4kT/(R·λ2), i.e. 4kT/(Req·λ). It results that, at equal C and time constant, the noise contribution from R is a factor λ lower than the one from R1 in the corresponding configuration of FIG. 2.
On the other hand, design constraints for linearity and dynamic range suggest that the dominant noise contribution comes from the channel noise of the last transistor of the feedback chain, MF.
We start analyzing the configuration in FIG. 6(a). This configuration can also make use of a cascode stage MC, as shown in the detail in FIG. 6(a), which is frequently used in complementary configurations as described in R. L. Chase, A. Hrisoho, and J. P. Richer, “8-channel CMOS preamplifier and shaper with adjustable peaking time and automatic pole-zero cancellation”, Nucl. Instrum. & Meth., A409 (1998) 328-331, and C. Fiorini and M. Porro, “Integrated RC cell for time-invariant shaping amplifiers”, IEEE Trans. Nucl. Sci., vol. 51, no. 5, pp. 1953-1960, 2004, each of which is incorporated by reference in its entirety as if fully set forth in this specification. It can be easily verified that, in order to guarantee a linear response, the relationship R·gmR>>1 must be satisfied, where gm is the transconductance of MR (or the one of a cascode MOSFET, if applicable). This relationship imposes a limit to the minimum current IR flowing through R. Assuming that MR operates in moderate inversion (as required to mirrors to guarantee a large enough voltage swing), its gm can be approximated as gmR≈IR/nVT, where n is the sub-threshold factor (n≈1.2 typical) and VT=kT/q is the thermal voltage (˜25 mV at 300 K). It follows the requirement on the voltage drop across R, given by R·IR>>nVT. Since MF operates in moderate inversion, its white noise spectral density is given by SnMF=2qIF. By considering the mirror ratio and by imposing the relationship for linear response it follows:
                                          S            nMF                    =                                    2              ⁢                              qI                F                                      =                                          2                ⁢                                  qI                  R                                            λ                                      >>                                            2              ⁢                              qnV                T                                                    R              ⁢                                                          ⁢              λ                                =                                    2              ⁢              kT                                      R                              eq                ⁢                                                                                                                          (        18        )            which shows that, at equal C and time constant (i.e., when Req=R1), the noise spectral density from MF would be larger than the one from R1 in FIG. 2, given by 4kT/R1. The low-frequency noise component from MF should also be added, but this contribution can be reduced, to some extent, by increasing the gate area of MF (i.e., by increasing both L and W of MF). The non-stationary noise contribution can be considered, which is due to the increase in the drain current of MF in presence of a signal. In the time domain this contribution, integrated in C, can be approximated as qiFτp/C2 where iF is the signal current and τp is the peaking time, which is a measure of the integration time. Additionally, the signal integrated in C is given by QAc/C, where Q is the input charge and Ac is the charge amplifier charge gain. By considering that iFτp≈QAc it follows for the signal-to-noise ratio due to the non-stationary contribution:
                                                        (                              S                N                            )                        ns                    ≈                                                    QA                c                            C                        ⁢                                                            C                  2                                                  qQA                  c                                                                    =                                                            Q                q                            ⁢                              A                c                                              =                                    NA              c                                                          (        19        )            where N is the number of signal electrons at the input of the charge amplifier. In most practical cases this contributions has negligible impact on the total S/N due to Ac>>1, as it can be observed assuming a minimum signal N≈ENC, which means ENC is the number of electrons. Attempts to improve the linearity by controlling the gate voltage of the cascode MC can be considered, as shown in FIG. 6(b) and in I. I. Jung, J. H. Lee, C. S. Lee, and Y. W. Choi, “Design of high-linear CMOS circuit using a constant transconductance method for gamma-ray spectroscopy system”, Nucl. Instrum. & Meth., A629 (2011) 277-281, which is incorporated by reference in its entirety as if fully set forth in this specification. However, the noise contribution from the controlling stages must be taken into account; also, maintaining the voltage drop across R below the thermal voltage VT might be challenging.
Most of the previous arguments apply to the configuration in FIG. 6(c) as described in G. Bertuccio, P. Gallina, and M. Sampietro, “‘R-Lens filter’: an (RC)n current-mode low-pass filter”, IEEE Electronics Letters, vol. 35, no. 15, pp. 1209-1210, 1999, which is incorporated by reference in its entirety as if fully set forth in this specification, where the MR is now the MOSFET used as source follower.
With regards to the configuration in FIG. 6(d), ideally the voltage drop across R1 can be kept small, but, in practical cases, it is difficult to reduce it to values much lower than the thermal voltage VT, and Equation (18) would still apply. A further challenge towards the various configurations in FIGS. 6(a)-(d) is to obtain a high linearity in the mirror stages over a wide dynamic range of currents.
The discussions above suggest that the linear configurations that make use of active devices in the signal path (e.g., current mirrors) cannot offer a dynamic range wider than the corresponding based on passive components only. It should be observed that OTA-based CMOS stages would enter this category as well, as described in T. Noulis, C. Deradonis, S. Siskos, and G. Sarrabayrouse, Nucl. Instrum. & Meth., A583 (2007) 469-478., which is incorporated by reference in its entirety as if fully set forth in this specification. The use of BiCMOS technologies would greatly alleviate the limitation in linearity, as shown in S. Buzzetti and C. Guazzoni, “A novel compact topology for high-resolution CMOS/BiCMOS spectroscopy amplifiers”, IEEE Trans. Nucl. Sci., vol. 52, no. 5, pp. 1611-1616, 2005, which is incorporated by reference in its entirety as if fully set forth in this specification, but some of the limitations previously discussed still apply, including the loss due to the voltage drops.
Therefore, there is a need for a low-noise analog shaper that provides high dynamic range.